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CHAPTER 2
2.2-1. The rectangular rules for numerical integration are illustrated in Fig. P2.2-1. The left-side rule is
depicted in Fig. P2.2-1(a), and the right-side rule is depicted in Fig. P2.2-1(b). The integral of x (t )
is approximated by the sum of the rectangular areas shown for each rule. Let y ( kT ) be the
numerical integral of x (t ), 0 ≤ t ≤ kT.
x(t)

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Y ( z)

X ( z)

=

(T 2)( z + 1)
z −1

Solution:
(a) y(k + 1) = y(k ) + T
(b) zY ( z) = Y ( z) +

x(k ) + x(k + 1)
2

T
T z +1
X ( z)
[ X ( z) + zX ( z)] ⇒ Y ( z) =
2
2 z −1

2.2-3. (a) The transfer function for the right-side rectangular-rule integrator was found in Problem 2.2-1
to be Y ( z ) /X ( z ) = Tz/ ( z − 1) . We would suspect that the reciprocal of this transfer function should
yield an approximation to a differentiator. That is, if w( kT ) is a numerical derivative of x (t ) at
t = kT ,

W ( z)
X ( z)

=

z −1
Tz

Write the difference equation describing this differentiator.
(b) Draw a figure similar to those in Fig. P2.2-1 illustrating the approximate differentiation.
(c) Repeat part (a) for the left-side rule, where W ( z ) /X ( z ) = T / ( z − 1) .
(d) Repeat part (b) for the differentiator of part (c).

Solution:
(a) Tz W ( z) = zX ( z) − X ( z)
w(k + 1) =

1
[ x(k + 1) − x(k )]
T

(b)
x
calculated
slope
kT (k + 1)T t

(c) TW ( z) = zX ( z) − X ( z)

19

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x

calculated
slope

kT (k + 1)T t

w(k ) =

1
[ x(k + 1) − x(k ) ]
T

2.3-1. Find the z-transform of the number sequence generated by sampling the time function e(t ) = t
every T seconds, beginning at t = 0 . Can you express this transform in closed form?

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(a) Find the conditions on the parameter a such that ⇥ ⎡⎢⎣cos akT ⎤⎥⎦ is first order (pole-zero cancellation
occurs).
(b) Give the first-order transfer function in part (a).
(c) Find a such that ⇥ ⎡⎢⎣cos akT ⎤⎦⎥ = ⇥ ⎡⎢⎣u ( kT )⎤⎥⎦ , where u ( kT ) is the unit step function.

2.5-2. Find the z-transform, in closed form, of the number sequence generated by sampling the time
function e(t ) every T seconds beginning at t = 0 . The function e(t ) is specified by its Laplace
transform,
E ( s) =

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(a) Solve for x ( k ) as a function of k.
(b) Evaluate x (0) , x (1) , x (2) , and x (3) in part (a).
(c) Verify the results in part (b) using the power-series method.
(d) Verify the results in part (b) by solving the difference equation directly.

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using the inversion formula.
(b) Check the value of e(0) using the initial-value property.
(c) Check the values calculated in part (a) using partial fractions.
30

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2

(e) Repeat parts (a) and (b) with E ( z ) = z/ ( z − 1.1) .

Solution:
(a) e(∞) = lim ( z − 1) E ( z ) =
z →1

z ( z − 1)
( z + 1) 2

=0
z =1

⎡ z ⎤
(b) e(k ) = z −1 ⎢
= k (−1)k , ∴ e(∞)unbounded
2⎥
⎣ ( z − 1) ⎦

(c) (a) e(∞) = lim ( z − 1)
z →1

z
, ∴ unbounded
( z − 1)2

(b) e(k ) = k , ∴ unbounded
(d) (a) e(∞) = lim ( z − 1)
z →1

z
=0
( z − 0.9)2

(b) e(k ) = k (0.9)k ; ∴ e(∞) → 0
(e) (a) e(∞) = lim ( z − 1)
z →1

z
=0
( z − 1.1)2

(b) e(k ) = k (1.1)k ; ∴ e(∞) is unbounded.

2.7-3. Find the inverse z-transform of each E ( z ) below by the four methods given in the text. Compare
the values of e( z ) , for k = 0, 1, 2, and 3, obtained by the four methods.
(a) E ( z ) =

(c) E ( z ) =

0.5z
( z − 1)( z − 0.6)

0.5( z + 1)

( z − 1)( z − 0.6)

(b)

E ( z) =

(d)

E ( z) =

0.5
( z − 1)( z − 0.6)

z ( z − 0.7)

( z − 1)( z − 0.6)

(e) Use MATLAB to verify the partial-fraction expansions.

Solution:

33

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E(z) = 1+ 0.9z −1 + 0.84z −2 + 0.804z −3 +!

(e) num=[0 0 0.5];
den=[1 –1.6 0.6];
[r, p, k] = residue (num, den)

2.8-1. Given in Fig. P2.8-1 are two digital-filter structures, or realizations, for second-order filters.
e(k)

T

T
d2

d1

+

d0

+

+

+

y(k)

y(k)

T

-

-

T

c1
c0
(a)
b2

b1

e(k)

+

f(k)
-

-

T

T

b0

+

+

y(k)

+

a1

a0
(b)

FIGURE P2.8-1 Digital-filter structures: (a) 3D; (b) 1D.
35

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(a) Write the difference equation for the 3D structure of Fig. P2.8-1(a), expressing y ( k ) as a
function of y ( k − i) and e( k − i) .
(b) Derive the filter transfer function Y ( z ) /E ( z ) for the 3D structure by taking the z-transform of
the equation in part (a).
(c) Write the difference equation for the 1D structure of Fig. P2.8-1(b). Two equations are
required, with one for f ( k ) and one for y ( k ) .
(d) Derive the filter transfer function Y ( z ) /E ( z ) for the 1D structure by taking the z-transform of
the equations in part (c) and eliminating F ( z ) .
(e) From parts (b) and (d), relate the coefficients α i , βi to ai , bi such that the two filters realize the
same transfer function.
(f) Write a computer-program segment that realizes the 3D structure. This program should be of
the form used in Example 2.10.
(g) Write a MATLAB-program segment that realizes the 1D structure. This program should be of
the form used in Example 2.10.

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(a) To realize this filter, difference equations are required for f1 ( k ), f2 ( k ), and y ( k ) . Write these
equations.
(b) Find the filter transfer function Y ( z ) /E ( z ) by taking the z-transform of the equations of part (a)
and eliminating F1 ( z ) and F2 ( z ) .
(c) Verify the results in part (b) using Mason’s gain formula.
(d) Write a MATLAB-program segment that realizes the 1X structure. This program should be of
the form of that is used in Example 2.10.
38

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(a) Find the coefficients of the 3D structure of Fig. P2.8-1 such that D ( z ) is realized.
(b) Find the coefficients of the ID structure of Fig. P2.8-1 such that D ( z ) is realized.
(c) Find the coefficients of the IX structure of Fig. P2.8-2 such that D ( z ) is realized.
The coefficients are identified in Problem 2.8-2.
(d) Use MATLAB to verify the partial-fraction expansions in part (c).
(e) Verify the results in part (c) using Mason’s gain formula.

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(a) Find the transfer function Y ( z ) /U ( z ) .
(b) Using any similarity transformation, find a different state model for this system.
(c) Find the transfer function of the system from the transformed state equations.
(d) Verify that A given and A w derived in part (b) satisfy the first three properties of similarity
transformations. The fourth property was verified in part (c).

Solution:
⎡ z −1 ⎤
⎥ ; Δ = zI − A = z ( z − 3) = Δ
⎣ 0 z − 3⎦

(a) zI − A = ⎢

44

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(a) Find d1 and d2 .
(b) Find a similarity transformation that results in the A w matrix given. Note that this matrix is
diagonal.
(c) Find B w and C w .
(d) Find the transfer functions of both sets of state equations to verify the results of this problem.

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(a) Find the transfer function Y ( z ) /U ( z ) .
(b) Using any similarity transformation, find a different state model for this system.
(c) Find the transfer function of the system from the transformed state equations.
(d) Verify that A given and A w derived in part (b) satisfy the first three properties of similarity
transformations. The fourth property was verified in part (c).

Solution:
(a)

47

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2.11-1. Consider a system with the transfer function
G ( z) =

Y ( z)

U ( z)

=

2
z ( z − 1)

(a) Find three different state-variable models of this system.
(b) Verify the transfer function of each state model in part (a), using (2-84).

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where u ( k ) is the system input.
(a) Find a state-variable formulation for this system. Consider the outputs to be y ( k + 1) and v ( k ) .
Hint: Draw a simulation diagram first.
(b) Repeat part (a) with y ( k ) and v ( k ) as the outputs.

50

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(c) Repeat part (a) with the single output v ( k ) .
(d) Use (2-84) to calculate the system transfer function with v ( k ) as the system output, as in part
(c); that is, find V ( z ) /U ( z ) .
(e) Verify the transfer function V ( z ) /U ( z ) in part (d) by taking the z-transform of the given system
difference equations and eliminating Y ( z ) .
(f) Verify the transfer function V ( z ) /U ( z ) in part (d) by using Mason’s gain formula on the
simulation diagram of part (a).

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(a) Calculate the transfer function Y ( z ) /U ( z ) , using (2-84).
(b) Draw a simulation diagram for this system, from the state equations given.
(c) Use Mason’s gain formula and the simulation diagram to verify the transfer function found in
part (a).

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generates a standard set of state equations for the transfer function whose numerator
coefficients are given in the vector num and denominator coefficients in the vector den.
(a) Use the MATLAB statement given to generate a set of state equations for the transfer function
G ( z) =

3z + 4
z + 5z + 6
2

(b) Draw a simulation diagram for the state equations in part (a).
(c) Determine if the simulation diagram in part (b) is one of the standard forms in Section 2.9.

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(a) Find the transfer function of this system.
(b) Let u ( k ) = 1, k ≥ 0 (a unit step function) and x (0) = 0 . Use the transfer function of part (a) to
find the system response.
(c) Find the state transition matrix Φ( k ) for this system.
(d) Use (2-90) to verify the step response calculated in part (b). This calculation results in the
response expressed as a summation. Then check the values y (0) , y (1) , and y (2) .
(e) Verify the results of part (d) by the iterative solution of the state equations.

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(a) Use (2-89) to solve for x ( k ), k ≥ 0.
(b) Find the output y ( z ) .
(c) Show that Φ( k ) in (a) satisfies the property Φ(0) = I.
(d) Show that the solution in part (a) satisfies the given initial conditions.
(e) Use an iterative solution of the state equations to show that the values y ( k ) , for k = 0, 1, 2, and
3, in part (b) are correct.
(f) Verify the results in part (e) using MATLAB.

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(a) Use (2-89) to solve for x ( k ), k ≥ 0.
(b) Find the output y ( k ) .
(c) Show that Φ( k ) in part (a) satisfies the property Φ(0) = I .
(d) Show that the solution in part (a) satisfies the given initial conditions.
(e) Use an iterative solution of the state equations to show that the values y ( k ) , for k = 0, 1, 2, and
3, in part (b) are correct.

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